- It is not the first time that Kupiansk residents have had to leave their homes.
- In March, children and people "with limited mobility" were ordered to evacuate from Kupiansk city because of an increase in Russian shelling.
Ukraine has ordered the mandatory evacuation of all civilians from 37 settlements in the north-east as Russia steps up its attacks there.
The authorities in the Kupiansk district of the Kharkiv region said they had to act because of "constant Russian shelling" in the area.
They said local residents were being moved to "safe regions" of Ukraine.
Russia says its troops have gained some ground in the area. Ukraine says Russian attacks have been rebuffed.
In a statement, the Kupiansk district authorities said residents of two towns and 35 villages were being evacuated.
"Do not neglect your safety and the safety of your loved ones!" the authorities said.
A resident in Kivsharivka - one of the villages being evacuated - said she was preparing to leave with her children, while her husband wanted to stay to care for his elderly mother.
"It's hard to leave them behind," Anna Koresh, 36, told AFP news agency. "But since it's getting dangerous it's important to take the kids to a safe place," she added.
In its latest briefing, the Russian defence ministry said its assault units in the Kupiansk direction had "improved their position along the front line during offensive operations".
Ukraine's military said its forces "successfully repelled" Russian attacks.
It is not the first time that Kupiansk residents have had to leave their homes.
In March, children and people "with limited mobility" were ordered to evacuate from Kupiansk city because of an increase in Russian shelling.
Kupiansk - an important transport and logistics regional hub - has witnessed fierce fighting since Russian President Vladimir Putin ordered a full-scale invasion of Ukraine on 24 February 2022.
Russian troops seized the city in a matter of days - but Ukrainian forces took back control during a rapid counter-attack last September.
Those advances - and the liberation of Ukraine's southern city of Kherson - were the most significant front-line changes since Russia withdrew from areas around the capital Kyiv in April.